Beginning Algebra With Applications, Chapter 5, 5.6, Section 5.6, Problem 12

Illustrate the solution set $\displaystyle x - y > - 3$


$
\begin{equation}
\begin{aligned}
x - y &> - 3
&& \text{Solve the inequality for } y\\
\\
-y &> -x - 3\\
\\
\frac{y}{-1} &< \frac{-x}{-1} - \frac{3}{-1}
&& \text{Remember that if you divide or multiply negative numbers ,the inequality symbol reverses}\\
\\
y &< x + 3
\end{aligned}
\end{equation}
$

To graph the inequality, we first find the intercepts of the line $\displaystyle y = x + 3$.
In this case, the $x$-intercept (set $y = 0$) is $\left( -3, 0 \right)$


$
\begin{equation}
\begin{aligned}
0 &= x + 3\\
\\
x &= -3
\end{aligned}
\end{equation}
$


And the $y$-intercept (set $x = 0$) is $(0, 3)$

$
\begin{equation}
\begin{aligned}
y &= 0 + 3\\
\\
y &= 3
\end{aligned}
\end{equation}
$

So the graph is


Graph is $y = x + 3 $ as a dashed line. Shade the lower half-plane.